Parametrized surface examples

Suggested background

Example 1

Parametrize the single cone $z=\sqrt{x^2+y^2}$.

Solution: For a fixed $z$, the cross section is a circle with
radius $z$. So, if $z=\spfv$, the parameterization of that circle is
$x=\spfv\cos \spsv$, $y=\spfv \sin \spsv$, for $0 \le \spsv \le 2\pi$.